When there are some points in momentum space give $|\nabla_k \varepsilon_k|=0$, they are called Van Hove points and give singularity in the desity of states (DOS). But what if $|\nabla_k \varepsilon_k|=0$ is zero along a line or in a surface.

For example, consider a dispersion relation in 1D like $$\varepsilon_k = \left\{ \begin{array}{ll} 0 & |k| \le 1 \\ |k|-1 & |k| > 1 \, . \end{array} \right.$$ What kind of divergence near $\epsilon=0$? It seems that the DOS for all $\varepsilon>0$ is a constant and suddenly goes to infinity when $\varepsilon$ is fine tuning to zero.

What about more general cases in 2D or 3D?

  • $\begingroup$ This is better. But the links at the end don't seem to add anything; I think you could safely take them out, and it would improve the question. I suggested linking to your other questions because you said they were closely related, so I thought you'd write them in such a way that it would provide useful context to links to the other parts, but actually each of these new questions is fairly self-contained; the links seem unnecessary. $\endgroup$ – David Z Jul 5 '15 at 12:32
  • $\begingroup$ Yeah...I rewrote those questions so that they are more self-contained as indenpendent posts. I'll take those links out. $\endgroup$ – Simon Jul 5 '15 at 12:40
  • $\begingroup$ Cool, that helps. Sorry, I didn't mean to imply you had to link the questions amongst each other (in case that was the impression you got). $\endgroup$ – David Z Jul 5 '15 at 12:52

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